separatrices

Separatrices are curves or trajectories in the phase space of a dynamical system that separate regions with qualitatively different motion. They often arise from saddle-type fixed points, where the stable and unstable manifolds form invariant curves that act as boundaries between distinct dynamical behaviors. A trajectory that lies on a separatrix marks the boundary between regions that converge to a fixed point and those that move away or behave differently over time. In many cases the separatrices are the stable and unstable manifolds of a saddle point.

In planar continuous-time systems, a common situation is a two-dimensional system dx/dt = f(x) with a saddle at a point p. The stable manifold W^s(p) consists of points that approach p as t → +∞, and the unstable manifold W^u(p) consists of points that depart from p as t → +∞. The separatrices are these invariant manifolds and form curves through p that divide the phase portrait into regions with distinct qualitative behavior. A simple example is a linear saddle, whose invariant lines serve as separatrices.

In Hamiltonian systems with two degrees of freedom, separatrices are energy level curves that pass through a saddle point and separate libration from rotation. The classic pendulum has a separatrix corresponding to the energy of the upright unstable equilibrium; crossing it changes the type of motion.

Global properties include closed separatrices (homoclinic) and connections between saddles (heteroclinic). Perturbations may destroy separatrices, producing complex dynamics and chaos through homoclinic tangles. The term originates from French and is widely used in dynamical systems theory and the study of differential equations.